Solved Exercises on Prime and Composite Numbers Practice in Grade 4

Master prime and composite numbers through practice: identifying primes, finding composites, understanding divisibility, and recognizing number properties through these 5 detailed exercises.

Practice Solutions: 1 to 3
1 Practice: Prime Identification
Practice Problem 1
Determine if 23 is a prime number. Show all your work and explain your reasoning.
Definition:

Prime Number: A number greater than 1 that has exactly two factors: 1 and itself

Method:
  1. Start with 1 and check if it divides the number evenly
  2. Check 2, 3, 4, 5, up to the number itself
  3. List all numbers that divide evenly
  4. Count the total factors
  5. If exactly 2 factors exist, it's prime
Number
23
Factors
1, 23
Count
2 factors
Step 1: Check divisibility by 1

23 ÷ 1 = 23 remainder 0 → 1 is a factor

Step 2: Check divisibility by 2

23 ÷ 2 = 11 remainder 1 → 2 is not a factor

Step 3: Check divisibility by 3

23 ÷ 3 = 7 remainder 2 → 3 is not a factor

Step 4: Check divisibility by 4

23 ÷ 4 = 5 remainder 3 → 4 is not a factor

Step 5: Check divisibility by 5

23 ÷ 5 = 4 remainder 3 → 5 is not a factor

Step 6: Check divisibility by 6

23 ÷ 6 = 3 remainder 5 → 6 is not a factor

Step 7: Check divisibility by 7

23 ÷ 7 = 3 remainder 2 → 7 is not a factor

Step 8: Check divisibility by 8

23 ÷ 8 = 2 remainder 7 → 8 is not a factor

Step 9: Check divisibility by 9

23 ÷ 9 = 2 remainder 5 → 9 is not a factor

Step 10: Check divisibility by 10

23 ÷ 10 = 2 remainder 3 → 10 is not a factor

Step 11: Check divisibility by 11

23 ÷ 11 = 2 remainder 1 → 11 is not a factor

Step 12: Check divisibility by 12

23 ÷ 12 = 1 remainder 11 → 12 is not a factor

Step 13: Check divisibility by 13

23 ÷ 13 = 1 remainder 10 → 13 is not a factor

Step 14: Check divisibility by 14

23 ÷ 14 = 1 remainder 9 → 14 is not a factor

Step 15: Check divisibility by 15

23 ÷ 15 = 1 remainder 8 → 15 is not a factor

Step 16: Check divisibility by 16

23 ÷ 16 = 1 remainder 7 → 16 is not a factor

Step 17: Check divisibility by 17

23 ÷ 17 = 1 remainder 6 → 17 is not a factor

Step 18: Check divisibility by 18

23 ÷ 18 = 1 remainder 5 → 18 is not a factor

Step 19: Check divisibility by 19

23 ÷ 19 = 1 remainder 4 → 19 is not a factor

Step 20: Check divisibility by 20

23 ÷ 20 = 1 remainder 3 → 20 is not a factor

Step 21: Check divisibility by 21

23 ÷ 21 = 1 remainder 2 → 21 is not a factor

Step 22: Check divisibility by 22

23 ÷ 22 = 1 remainder 1 → 22 is not a factor

Step 23: Check divisibility by 23

23 ÷ 23 = 1 remainder 0 → 23 is a factor

Factors of 23: 1, 23 (exactly 2 factors)
Final answer:

Yes, 23 is a prime number because it has exactly two factors: 1 and 23.

Applied rules:

• Prime Definition: Must have exactly two factors

• Factor Testing: Check divisibility systematically

• Counting: Only 2 factors means it's prime

2 Practice: Composite Identification
Practice Problem 2
Is 30 a composite number? Find all its factors and explain why.
Definition:

Composite Number: A number greater than 1 that has more than two factors

Number
30
Factors
1, 2, 3, 5, 6, 10, 15, 30
Count
8 factors
Step 1: Check divisibility by 1

30 ÷ 1 = 30 remainder 0 → 1 is a factor

Step 2: Check divisibility by 2

30 ÷ 2 = 15 remainder 0 → 2 is a factor

Step 3: Check divisibility by 3

30 ÷ 3 = 10 remainder 0 → 3 is a factor

Step 4: Check divisibility by 4

30 ÷ 4 = 7 remainder 2 → 4 is not a factor

Step 5: Check divisibility by 5

30 ÷ 5 = 6 remainder 0 → 5 is a factor

Step 6: Check divisibility by 6

30 ÷ 6 = 5 remainder 0 → 6 is a factor

Step 7: Check divisibility by 7

30 ÷ 7 = 4 remainder 2 → 7 is not a factor

Step 8: Check divisibility by 8

30 ÷ 8 = 3 remainder 6 → 8 is not a factor

Step 9: Check divisibility by 9

30 ÷ 9 = 3 remainder 3 → 9 is not a factor

Step 10: Check divisibility by 10

30 ÷ 10 = 3 remainder 0 → 10 is a factor

Step 11: Check divisibility by 11

30 ÷ 11 = 2 remainder 8 → 11 is not a factor

Step 12: Check divisibility by 12

30 ÷ 12 = 2 remainder 6 → 12 is not a factor

Step 13: Check divisibility by 13

30 ÷ 13 = 2 remainder 4 → 13 is not a factor

Step 14: Check divisibility by 14

30 ÷ 14 = 2 remainder 2 → 14 is not a factor

Step 15: Check divisibility by 15

30 ÷ 15 = 2 remainder 0 → 15 is a factor

Step 16: Check divisibility by 16

30 ÷ 16 = 1 remainder 14 → 16 is not a factor

Step 17: Check divisibility by 17

30 ÷ 17 = 1 remainder 13 → 17 is not a factor

Step 18: Check divisibility by 18

30 ÷ 18 = 1 remainder 12 → 18 is not a factor

Step 19: Check divisibility by 19

30 ÷ 19 = 1 remainder 11 → 19 is not a factor

Step 20: Check divisibility by 20

30 ÷ 20 = 1 remainder 10 → 20 is not a factor

Step 21: Check divisibility by 21

30 ÷ 21 = 1 remainder 9 → 21 is not a factor

Step 22: Check divisibility by 22

30 ÷ 22 = 1 remainder 8 → 22 is not a factor

Step 23: Check divisibility by 23

30 ÷ 23 = 1 remainder 7 → 23 is not a factor

Step 24: Check divisibility by 24

30 ÷ 24 = 1 remainder 6 → 24 is not a factor

Step 25: Check divisibility by 25

30 ÷ 25 = 1 remainder 5 → 25 is not a factor

Step 26: Check divisibility by 26

30 ÷ 26 = 1 remainder 4 → 26 is not a factor

Step 27: Check divisibility by 27

30 ÷ 27 = 1 remainder 3 → 27 is not a factor

Step 28: Check divisibility by 28

30 ÷ 28 = 1 remainder 2 → 28 is not a factor

Step 29: Check divisibility by 29

30 ÷ 29 = 1 remainder 1 → 29 is not a factor

Step 30: Check divisibility by 30

30 ÷ 30 = 1 remainder 0 → 30 is a factor

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 (8 factors total)
Final answer:

Yes, 30 is a composite number because it has more than two factors: 1, 2, 3, 5, 6, 10, 15, and 30.

Applied rules:

• Composite Definition: Must have more than two factors

• Factor Pairs: 1×30=30, 2×15=30, 3×10=30, 5×6=30

• Counting: More than 2 factors means it's composite

3 Practice: Classification
Practice Problem 3
From the list below, classify each number as prime or composite: 19, 21, 25, 29, 31
Definition:

Classification: Determining if each number is prime or composite by finding its factors

Numbers
19, 21, 25, 29, 31
Factors
List each
Types
P, C, C, P, P
Step 1: Analyze 19

Factors: 1, 19 → Prime (exactly 2 factors)

Step 2: Analyze 21

Factors: 1, 3, 7, 21 → Composite (4 factors)

Step 3: Analyze 25

Factors: 1, 5, 25 → Composite (3 factors)

Step 4: Analyze 29

Factors: 1, 29 → Prime (exactly 2 factors)

Step 5: Analyze 31

Factors: 1, 31 → Prime (exactly 2 factors)

Step 6: Final classification

Prime: 19, 29, 31 | Composite: 21, 25

Prime: 19, 29, 31 | Composite: 21, 25
Final answer:

Prime numbers: 19, 29, 31 | Composite numbers: 21, 25

Applied rules:

• Systematic Analysis: Examine each number individually

• Factor Counting: Determine the exact number of factors

• Classification Rules: 2 factors = prime, more than 2 = composite

Practice Concepts & Methods
Prime: 2 Factors | Composite: >2 Factors
Classification Rule
Rule 1
Prime: exactly 2 factors
Prime Definition
Rule 2
Composite: more than 2 factors
Composite Definition
Rule 3
Special: 1 = neither prime nor composite
Special Case
Key definitions:

Prime Number: A number greater than 1 that has exactly two factors: 1 and itself

Composite Number: A number greater than 1 that has more than two factors

Factors: Numbers that divide evenly into another number without leaving a remainder

Practice methodology:
  1. Check if number > 1: Prime/composite only applies to numbers greater than 1
  2. Find all factors: Systematically check divisibility from 1 up to the number
  3. Count factors: Tally the total number of factors found
  4. Classify: 2 factors = prime, more than 2 = composite
Tip 1: Even numbers (except 2) are always composite since they're divisible by 2.
Tip 2: Numbers ending in 5 (except 5) are composite since they're divisible by 5.
Tip 3: The number 2 is the only even prime number.
Tip 4: To save time, only test divisors up to the square root of the number.
Common errors: Thinking 1 is prime, missing factors when counting, confusing factors with multiples.
Key concepts: Prime numbers are the building blocks of all numbers, every composite number can be broken down into primes.
Rules to remember:

• 1 is neither prime nor composite

• 2 is the smallest and only even prime number

• All primes greater than 2 are odd numbers

• The first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...

• Every composite number has at least one prime factor

Practice Solutions: 4 to 5
4 Practice: Word Problem
Practice Problem 4
Lisa has 37 stickers. She wants to distribute them equally among some friends with no stickers left over. How many friends can she share with? Is 37 prime or composite?
Definition:

Word problems: Real-world situations requiring knowledge of factors and divisibility

Sticker count
37
Factors
1, 37
Sharing options
1 friend or 37 friends
Step 1: Find factors of 37

Factors of 37: 1, 37

Step 2: Interpret sharing options

Option 1: Share with 1 friend (37 ÷ 1 = 37 stickers each)

Option 2: Share with 37 friends (37 ÷ 37 = 1 sticker each)

Step 3: Determine if 37 is prime or composite

Since 37 has exactly 2 factors (1 and 37), it's prime

Step 4: Answer the question

Lisa can share with 1 friend (giving 37 stickers) or 37 friends (giving 1 sticker each)

37 is prime, so Lisa can share with 1 or 37 friends
Final answer:

Lisa can share her 37 stickers with 1 friend (37 stickers each) or 37 friends (1 sticker each). The number 37 is prime because it has exactly two factors: 1 and 37.

Applied rules:

• Prime Application: Prime numbers have limited sharing options

• Real-world connection: Equal sharing requires understanding factors

• Verification: Confirm that sharing uses all items

5 Practice: Pattern Recognition
Practice Problem 5
Find all prime numbers between 40 and 50. List them and explain your reasoning.
Definition:

Range analysis: Systematically checking numbers within a specified range

Range
40-50
Check each
41, 42, 43, 44, 45, 46, 47, 48, 49
Primes
41, 43, 47
Step 1: List numbers 41-49 (excluding 40 and 50)

41, 42, 43, 44, 45, 46, 47, 48, 49

Step 2: Check 41

Factors: 1, 41 → prime

Step 3: Check 42

42 = 2 × 21 → composite

Step 4: Check 43

Factors: 1, 43 → prime

Step 5: Check 44

44 = 4 × 11 → composite

Step 6: Check 45

45 = 5 × 9 → composite

Step 7: Check 46

46 = 2 × 23 → composite

Step 8: Check 47

Factors: 1, 47 → prime

Step 9: Check 48

48 = 6 × 8 → composite

Step 10: Check 49

49 = 7 × 7 → composite

Prime numbers between 40 and 50: 41, 43, 47
Final answer:

The prime numbers between 40 and 50 are: 41, 43, and 47

Applied rules:

• Systematic approach: Check each number in the range individually

• Even number rule: All even numbers > 2 are composite

• Verification: Double-check each classification

Practice Concepts and Key Ideas
Prime: 2 Factors | Composite: >2 Factors
Practice Classification
Practice definitions:

Prime Number Practice: Exercises focusing on identifying numbers with exactly two factors

Composite Number Practice: Problems focusing on identifying numbers with more than two factors

Classification Practice: Activities that require distinguishing between prime and composite numbers

Practice approach:
  1. Practice 1 (Prime): Take number 17 → factors are 1 and 17 → exactly 2 factors → prime
  2. Practice 2 (Composite): Take number 24 → factors are 1, 2, 3, 4, 6, 8, 12, 24 → 8 factors → composite
  3. Practice 3 (Pattern): Notice that all even numbers > 2 are composite
  4. Practice 4 (Application): Prime numbers limit how items can be shared equally
Practice Tip 1: The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
Practice Tip 2: All primes > 2 are odd numbers (since even numbers > 2 are divisible by 2)

Practice Tip 3: Primes become less frequent as numbers get larger

Practice Tip 4: Every composite number can be broken down into prime factors

Practice Properties: Prime numbers are like building blocks - everything else is made from them. Composite numbers are like "group projects" made up of smaller prime numbers.

Real-world Practice: Computer security uses large prime numbers to keep passwords and messages safe.

Practice rules:

• 2 is the only even prime number

• All primes > 2 are odd numbers

• Numbers ending in 5 (except 5) are composite

• Every number > 1 is either prime or composite

• Prime factorization is unique for each number

Questions & Answers

Question: Why is 1 not a prime number? It seems like it should be since it only divides by itself.

Answer: This is a great question that confuses many students!

The definition of a prime number specifically states it must have exactly two different factors: 1 and itself. Since 1 only has one factor (itself), it doesn't meet this requirement.

Think of it like this: prime numbers are special because they can only be arranged in exactly two rectangular ways (1 row or 1 column). The number 1 can only be arranged in one way, so it gets its own special category.

So 1 is neither prime nor composite - it's just special by itself!

Question: How can I quickly tell if a number is prime without checking every single number?

Answer: You can use several smart strategies to make this easier!

  • Check divisibility by small primes first: 2, 3, 5, 7, 11
  • Only check up to the square root: If n is your number, only test divisors up to √n
  • Use divisibility rules: If the sum of digits is divisible by 3, the number is divisible by 3

For example, to check if 47 is prime:

  • It's not even, so not divisible by 2
  • Sum of digits: 4+7=11, not divisible by 3
  • Doesn't end in 0 or 5, so not divisible by 5
  • √47 ≈ 6.9, so we only check up to 6
  • Check 7: 47 ÷ 7 = 6 remainder 5 → not divisible

Since we've checked all primes up to √47 and none divide 47, it's prime!

Question: Are there any patterns to help me remember which numbers are prime?

Answer: Yes, there are helpful patterns to recognize prime numbers!

  • All primes > 2 are odd (except 2, which is the only even prime)
  • Primes > 5 end in 1, 3, 7, or 9 (never 0, 2, 4, 5, 6, 8)
  • Primes become less frequent as numbers get larger

The first 15 primes to remember: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Notice that after 2 and 5, all primes end in 1, 3, 7, or 9!